You have found the following ages (in years) of all 4 gorillas at your local zoo: $ 1,\enspace 5,\enspace 5,\enspace 8$ What is the average age of the gorillas at your zoo? What is the variance? You may round your answers to the nearest tenth.
Answer: Because we have data for all 4 gorillas at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $4$ ages and divide by $4$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\mu} = \dfrac{1 + 5 + 5 + 8}{{4}} = {4.8\text{ years old}} $ Find the squared deviations from the mean for each gorilla. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $1$ year $-3.8$ years $14.44$ years $^2$ $5$ years $0.2$ years $0.04$ years $^2$ $5$ years $0.2$ years $0.04$ years $^2$ $8$ years $3.2$ years $10.24$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{14.44} + {0.04} + {0.04} + {10.24}} {{4}} $ $ {\sigma^2} = \dfrac{{24.76}}{{4}} = {6.19\text{ years}^2} $ The average gorilla at the zoo is 4.8 years old. The population variance is 6.19 years $^2$.